What does this mean,(A Δ B)^c?

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In summary, (A Δ B)^c refers to the complement of the symmetric difference between sets A and B, which is represented by (A Δ B) = (A-B) U (B-A). This means that (A Δ B)^c would include all elements that are not in the symmetric difference of A and B. To find the complement, you would need to look at all elements in the universe that are not in (A Δ B). This can be written as (A Δ B)^c = {x | x ∉ (A Δ B)}.
  • #1
matrix_204
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What does this mean,

(A Δ B)^c? And what does it equal to?

I know that (A Δ B)= (A-B) U (B-A).
 
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  • #2
matrix_204 said:
What does this mean,

(A Δ B)^c? And what does it equal to?

I know that (A Δ B)= (A-B) U (B-A).

Might mean the complement of (A Δ B)
 
  • #3
Yes, I know, but how do I find the complement. Thats where I'm stuck. What is the complement equal to?
 
  • #4
matrix_204 said:
Yes, I know, but how do I find the complement. Thats where I'm stuck. What is the complement equal to?

I don't really understand what you are asking, the complement is going to be all members of the universe that are not in the given set so from what you have told me it would be the set of all x such that x is not a member of the symmetric difference between A and B.
 
  • #5
So for example, all x such that x is not in (A Δ B)= (A-B) U (B-A)!
What I was asking is what's (A Δ B)^c= ??
Because in the textbook it's not given and I got stuck for one of the problems before I could proceed.
 
  • #6
Note that your triangle thingy set does not contain points common to A and B, nor does it contain points in the complement of the union of A and B.

Does that help?
 

FAQ: What does this mean,(A Δ B)^c?

What is the meaning of (A Δ B)^c?

The notation (A Δ B)^c represents the complement of the symmetric difference between sets A and B. This means it includes all elements that are not in the symmetric difference of A and B.

How is (A Δ B)^c different from (A ∩ B)^c?

The notation (A ∩ B)^c represents the complement of the intersection between sets A and B. This means it includes all elements that are not in both A and B. In contrast, (A Δ B)^c includes all elements that are not in the symmetric difference of A and B, which includes elements that are in both A and B.

Can you provide an example of (A Δ B)^c?

Let A = {1, 2, 3} and B = {3, 4, 5}. The symmetric difference of A and B is (A Δ B) = {1, 2, 4, 5}. Therefore, (A Δ B)^c = {3}, since it includes all elements that are not in the symmetric difference of A and B.

How can (A Δ B)^c be represented using other set operations?

(A Δ B)^c = (A ∪ B) ∩ (A ∩ B)^c

In other words, it is the intersection between the union of A and B, and the complement of the intersection between A and B.

What are some real-world applications of (A Δ B)^c?

The concept of (A Δ B)^c is often used in probability and statistics, where it represents the complement of the event A and B both occurring. It can also be used in database operations to find all the elements that are not common between two sets of data.

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